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An equilibrium in a discrete logistic model represents a population size that remains constant over time. In mathematical terms, this is where the change in population from one period to the next is zero. For the given model, this means finding values of population size, denoted by \(N^*\), such that \(N_{t+1} = N_t = N^*\).

Insert the equilibrium condition into the equation: \\[ N^* = R_0 N^* - \frac{1}{100} (N^*)^2 \] Simplifying leads to: \\[ 0 = (R_0 - 1) N^* - \frac{1}{100} (N^*)^2 \] Factor this equation to find: \\[ N^*(R_0 - 1 - \frac{1}{100}N^*) = 0 \] This indicates two potential equilibria: one where \(N^* = 0\), meaning the population becomes extinct, and another found at \(N^* = 100(R_0 - 1)\), based on the growth rate \(R_0\).

Stability analysis helps us determine whether a population will remain around an equilibrium over time or deviate from it. This is crucial for understanding long-term population behavior.

To assess stability, we examine the derivative \(f'(N)\) of the function \(f(N) = R_0 N - \frac{1}{100} N^2\). Derive: \[ f'(N) = R_0 - \frac{2}{100} N \] Evaluate \(f'(N)\) at the equilibria:

• At \(N^* = 0\), the derivative is \(f'(0) = 3.5\). Since \(|f'(0)| > 1\), this equilibrium is unstable.
• At \(N^* = 250\), substitute into the derivative to find \(f'(250) = 3.5 - \frac{500}{100} = -1.5\). As \(|f'(250)| < 1\), this equilibrium is stable.

The stable equilibrium indicates that the population will tend to stabilize around \(N^* = 250\) in the long run.

Population dynamics in a discrete logistic model reveal patterns of growth and behavior over time. For the given exercise, examining how the population evolves allows us to understand its response to initial conditions and its path toward equilibrium.

Starting with an initial population \(N_0\) such as 10, we calculate subsequent population sizes using the recurrence relation:\( N_{t+1} = R_0 N_t - \frac{1}{100} N_t^2 \). The first ten terms give us insights into the dynamics:

• Rapid Growth: Initially, the population grows quickly beyond the unstable equilibrium of \(N^* = 0\).
• Fluctuation and Stabilization: The population continues to rise and will fluctuate as it nears the stable equilibrium \(N^* = 250\).
• Long-term Stability: Eventually, it stabilizes at or around the stable point, assuming conditions remain consistent.

This analysis highlights how populations may grow beyond limits and self-correct, a behavior typical of logistic models.

Recurrence relations are mathematical expressions that show how a sequence evolves based on its previous terms. In the context of population models, they are crucial for representing consecutive changes in population over discrete time.

For our logistic model, the recurrence relation is: \\[ N_{t+1} = R_0 N_t - \frac{1}{100} N_t^2 \] This equation helps track how each population size (\(N_{t+1}\)) depends directly on the previous size (\(N_t\)) and other parameters like the intrinsic growth rate \(R_0\). Using a recurrence relation:

• Enables predictions about future population size, given an initial population \(N_0\).
• Allows for observing how adjustments in \(R_0\) affect overall growth and approach to equilibrium.
• Models the biological constraints perfectly, especially when considering carrying capacity limits.

Hence, recurrence relations are a versatile tool in making informed iterations about natural processes like population growth.